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Full text of "Mathematical And Physical Papers - Iii"

ON THE  CONDUCTION  OF  HEAT  IN CRYSTALS.             215
If the crystal be covered by a thin coating of a given substance, h will be constant, and k will be a function of I, m, n, or of X, A6, v} which is determined by (14) and (16). These formulae shew that in the case supposed k will have the same value for the opposite faces of a plate bounded by parallel surfaces.
By means of the auxiliary solid, we may reduce problems relating to the conduction of heat in crystals to corresponding problems relating to ordinary media; or, conversely, from a set of self-evident or known results relating to ordinary media, we may deduce a set of corresponding results relating to crystals.
12. Let us first regard the crystal as infinite, in which case the auxiliary solid will be infinite likewise.
In an ordinary medium, if heat be introduced at one point according to any law, the isothermal surfaces will be a system of spheres, having the source of heat for their common centre, and the flow of heat at any point will take place in the direction of the radius vector drawn from the source. If the temperature be permanent, and the temperature at an infinite distance vanish, the temperature at any point will vary inversely as the distance from the source.
Hence, in a crystal, if heat be introduced at one point according to any law, the isothermal surfaces will be a system of similar and concentric ellipsoids, having their principal axes in the direction of the thermic axes drawn through the source, and proportional to the square roots of the principal conductivities. The flow of heat at any point will take place in the direction of the radius vector drawn from the source. If the temperature be permanent, and vanish at an infinite distance, the temperature along a given radius vector will vary inversely as the distance from the source.
It will frequently be convenient to refer to an ellipsoid constructed with its principal axes in the direction of the thermic axes, and equal to ^\/A} 2\/tt, 2 V^-s respectively. I shall call this ellipsoid the thermic
13. In an ordinary medium, whether finite or infinite, in which the temperature varies from point to point, and may be either constant or variable as regards the time, the flow of heat at any point takes place in the direction of the normal to the